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BUCKLING AND POST BUCKLING OF

STRUCTURAL COMPONENTS

by

SHRUTI DESHPANDE

Presented to the Faculty of the Graduate School of

The University of Texas at Arlington in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

THE UNIVERSITY OF TEXAS AT ARLINGTON

December 2010


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Copyright by Shruti Deshpande 2010

All Rights Reserved


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ACKNOWLEDGEMENTS

I am sincerely grateful to my thesis advisor Dr. Kent Lawrence, for his valuable

guidance and support throughout this thesis. He has consistently helped me improve and

supported me to gain my accomplishments. The requirements of this thesis were achieved in a

timely and organized manner under his guidance. Without him the entire work would be a

reckless venture.

I would like to thank Dr. B. P. Wang and Dr. Seiichi Nomura for taking the time to serve

on my thesis committee.

I am indebted to my family and especially to my husband Chinmay for his continuous

support and encouragement in all my endeavors. My Masters degree wouldnt have been

possible without him. I also thank all my friends, especially my roommates Athena and Rasika

who were a great source of encouragement.

November 23, 2010


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ABSTRACT

BUCKLING AND POST BUCKLING OF

STRUCTURAL COMPONENTS

Shruti Deshpande, M. S.

The University of Texas at Arlington, 2010

Supervising Professor: Kent Lawrence

Structural members and components, such as shells and trusses of different

geometries, form the intricate and deep seated parts in the manufacturing of missiles

submarines, rockets, airplanes, automobiles etc and find applications in civil structures such as

bridges. These members are comprised of components like conical frusta, cylindrical panels

and shallow trusses. In many cases, the sole purpose of such shell structures is to absorb the

energy generated due to impact which means, that these structures are subjected to heavy

loads and can experience failure due to buckling.

The objective of this work is to study the buckling and post buckling behavior of such

members. The study is carried out using finite element analysis. The widely implemented

softwares ANSYS APDL and ANSYS Workbench are used to perform the analysis.

The components analyzed consist of shell structures such as conical frusta and

cylindrical panels, and other structures like the shallow truss, diagonal truss and the shallow

arch. These structures are analyzed for their buckling and post buckling behavior when subject

to specific loading conditions and geometric, contact and material non-linearity. The results

compare favorably with known solutions.


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TABLE OF CONTENTS

ACKNOWLEDGEMENTS ....................................................................................................... iii

ABSTRACT ............................................................................................................................. iv

LIST OF ILLUSTRATIONS..................................................................................................... vii

LIST OF TABLES .................................................................................................................... ix

Chapter Page

1. INTRODUCTION...... .............................1

1.1 Scope of Study.........................................................................................2

2. LITERATURE REVIEW..........................................................................................4

3. BUCKLING AND POST BUCKLING ......................................................................8

3.1 Buckling....................................................................................................8

3.2 Post Buckling............................................................................................8

3.2.1 Geometric Non linearity............................................................9

3.2.2 Material Non linearity .............................................................10

3.2.3 Contact Non linearity..............................................................10

3.3 Finite Element Method and ANSYS.......................................................10

4. TRUSS AND ARCH STRUCTURES....................................................................12

4.1 Investigation Process .............................................................................12

4.2 ANSYS Simulation .................................................................................12

4.2.1 Shallow Truss.........................................................................13

4.2.1.1 ANSYS Simulation ..........................................13

4.2.1.2 Results............................................................17

4.2.2 Diagonal Truss .......................................................................22



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4.2.2.2 Results.............................................................25

4.2.3 Shallow Arch ..........................................................................27


4.2.3.2 Results............................................................29

5. SHELL STRUCTURES ........................................................................................32

5.1 Cylindrical Panel .......................................................................32

5.1.1 ANSYS Simulation .............................................32

5.1.2 Results...............................................................35

5.2 Conical Frustum ........................................................................38

5.2.1 ANSYS Simulation .............................................38

5.2.2 Results...............................................................41

5.3 Eigen Value Buckling Load .......................................................44

6. CONCLUSIONS AND RECOMMENDATIONS....................................................46

APPENDIX A. DEFORMED SPECIMES OF CONICAL FRUSTA............................47

APPENDIX B. EIGEN VALUE BUCKLING MODE SHAPES....................................50

REFERENCES.......................................................................................................................54

BIOGRAPHICAL INFORMATION ..........................................................................................56


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LIST OF ILLUSTRATIONS

Figure Page

4.1 Shallow Truss geometry...........................................................................................................13

4.2 BEAM3 element .......................................................................................................................14

4.3 Clamped Truss subjected to Boundary conditions...................................................................15

4.4 LINK1 element.......................................................................................................................... 16

4.5 Truss subjected to hinged boundary conditions.......................................................................16

4.6 Deformed clamped truss. .........................................................................................................17

4.7 Load (lb) Vs displacement (in). ................................................................................................18

4.8 Load (lb) Vs Horizontal reaction...............................................................................................19

4.9 Deformed hinged truss.............................................................................................................20

4.10 Load (lb) Vs deflection (in). ....................................................................................................21

4.11 Load (lb) Vs Horizontal Reaction (lb). ...................................................................................22

4.12 Diagonal truss Geometry. .....................................................................................................23

4.13 Stress Strain Curve. ..............................................................................................................24

4.14 Boundary conditions for Diagonal Truss. ..............................................................................25

4.15 Deformed Shape of Diagonal Truss.......................................................................................26

4.16 Load (lb) Vs Deflection (in).....................................................................................................27

4.17 Shallow Arch geometry. .........................................................................................................28

4.18 Boundary Conditions for Shallow Arch...................................................................................29

4.19 Deformed Shape of Shallow Arch with central displacement = -1.8 in. .................................30

4.20 Deformed Shape of Shallow Arch with central displacement = -0.5 in. .................................30

4.21 Load (lb) Vs Displacement (in)...............................................................................................31

5.1 Element Shell 181. ...................................................................................................................32


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5.2 Geometry of Cylindrical Panel..................................................................................................33

5.3 Mesh of Cylindrical Panel.........................................................................................................33

5.4 Applied Displacement at central Node. ....................................................................................34

5.5 Applied Force at Central node. ................................................................................................35

5.6 Deformed Shape of Cylindrical Panel . ...............................................................................35

5.7 Force Reaction at Central Node...............................................................................................36

5.8 Load Vs Displacement. ............................................................................................................37

5.9 Deformed Shape of panel due to applied load.........................................................................38

5.10 Geometry of Frustum. ............................................................................................................39

5.11 Mesh and geometry of specimen C6. ....................................................................................40

5.12 Boundary conditions applied to specimen C6........................................................................41

5.13 Deformed Shape of frustum...................................................................................................42

5.14 Reaction Force evaluated at contact region. .........................................................................42

5.15 Load Vs Displacement. ..........................................................................................................43

A.1 Example of Mesh density for frusta in reference [2]. ...............................................................48

A.2 C25 Deformed Shape for thickness of 0.7 mm........................................................................48

A.3 C18 Deformed Shape. .............................................................................................................49

A.4 C2 Deformed Shape. ...............................................................................................................49

B.1 Shallow Truss - Clamped.........................................................................................................51

B.2 Shallow Truss - Hinged............................................................................................................51

B.3 Diagonal Truss.........................................................................................................................52

B.4 Shallow Arch. ...........................................................................................................................52

B.5 Cylindrical Panel. .....................................................................................................................53

B.6 Conical Frustum Specimen C6. ...............................................................................................53


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LIST OF TABLES

Table Page

4.1 Dimensions of Shallow Truss...................................................................................................14

4.2 Material Properties for Shallow Truss. .....................................................................................14

4.3 Diagonal Truss Dimensions. ....................................................................................................23

4.4 Material Properties for Diagonal Truss.....................................................................................24

4.5 Dimensions of Shallow Arch. ...................................................................................................27

4.6 Material Properties of Shallow Arch. ........................................................................................28

5.1 Dimensions of Panel. ...............................................................................................................32

5.2 Material Properties of panel. ....................................................................................................34

5.3 Dimensions of Conical frusta. ..................................................................................................40

5.4 Material Properties of Frusta....................................................................................................41

5.5 Material Properties of Rigid Plate.............................................................................................42

5.6 Force Values for Frusta............................................................................................................44

5.7 Eigen Value Buckling Load compared to the Non-linear Buckling Limit Load.........................45


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CHAPTER 1

INTRODUCTION

The current manufacturing industry emphasizes on the use of structural members,

which encompass light weight with high strength and are expected to absorb high energy and

carry heavy loads. These members are comprised of Shell structures and shallow trusses.

The shell structures have a significant advantage in regards to load carrying capacity.

When load is applied to a shell it results into different types of internal forces which can cause

bending, twisting, transverse shearing and buckling. Commonly used components are

cylindrical panels, cylindrical and spherical shells and conical frusta. In many cases, these shell

structures are the deep seated structures in manufacturing submarines, missiles, tanks and

their roofs and fluid reservoirs. These components act as energy absorbers in most of these

cases.

These energy absorbing devices may be classified on the basis of the following

principles,

- extrusion

- friction

- material deformation

Shell structures such as cylindrical tubes and conical frusta are the most frequently

used devices that absorb energy by material deformation. Such axis-symmetric shell structures

are widely implemented in aircraft and spacecraft structures, nuclear reactors, storage tanks for

bulk solids and liquids, pressure vessels, pipelines etc.. Conical shells also find applications in

marine engineering. These are mainly implemented as transition elements beween cylinders of

different diameters. Shell structures also find versatile applications in the field of civil


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engineering. Some examples that have thin shells as structural components are silos, roofs,

container, tanks, pipes, pressure vessels, submarines and aircraft wings.

Structural Members such as Shallow Truss and arch find many applications in the field

of civil engineering. These form an important part of various structures like air plane hangars,

bridges, railways, roof structures etc. Arches are used for underground structures such as

drains and vaults and are widely used structural members in bridges and buildings too. The

arch can carry a much greater load thus making it more significant.

Buckling of these kind of structures is dependent on various number of variables like

geometry, material properties and the applied load. The Finite element methods (FEM)

technique is extensively used to analyze the buckling and post bucking behavior of such

structural components. Several finite element softwares like ADINA, NASTRAN, ANSYS and

ABAQUS are implemented for buckling and post bucking analysis.

Several papers that study the theories of the collapse of shell and truss structures have

appeared in the literature. The buckling analysis of such structures using finite element methods

occurs in the literature.

1.2 Scope of Study

The focus of this thesis is the study of buckling and post buckling of structural

components using ANSYS APDL and ANSYS Workbench. The thesis will discuss a number of

components and structural members such as The Shallow Arch, Shallow Truss, Diagonal Truss,

Cylindrical Panel and Conical Frusta.

The entire work area comprised of a number of chapters. For conducting the analysis

an intense literature review on the buckling analysis of thin shells like cylindrical shells, conical

shells, cylindrical panels and trusses is conducted, which is briefed in the second chapter. A

brief review of the buckling and post buckling concept is presented in the third chapter. The

fourth chapter details about the analysis of Shallow truss, Diagonal truss and the shallow arch

for buckling and post buckling behavior using a finite element methods software, ANSYS APDL.

The fifth chapter details about the buckling and post buckling analysis of the shells, such as the


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cylindrical panel and the conical frustum using the finite element methods software ANSYS

Workbench. A study of Eigen value buckling for all the cases is also included in the fifth chapter.

This is followed by conclusion and recommendation in the sixth chapter and then the references

used for the entire achievement.


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stress strain curve obtained from the experiments. The results thus obtained were higher but

could be compared well with the experimental ones. Using finite elements modeling thus

facilitates the analysis of intermediate stages of buckling which reduces the cost and time.

A.Pica and R. D. Wood have studied the post buckling behavior of plates and shells

using a mindlin shallow shell formulation[3]. This paper presents a geometrically non-linear

analysis of shallow shells using finite element mindlin formulation. It gives results for post

buckling behavior of square and circular plates subject to direct in plane loading and square

plate subject to in plane shear loading. Analysis of shallow truss and cylindrical and spherical

shells are also presented, all exhibiting snap through behavior. For a number of post buckling

solutions the 9 node lagrangian element was used which demonstrates the ability to model

curved boundaries.

Chawalit Thinvongpituk and Pisit Techarungpaisarn have studied the Buckling of Axially

compressed conical shells of linearly variable thickness using structural model[4]. The study

was conducted with a series of experiments performed using conical specimens with constant

thickness, which were crushed till the buckling load was recorded. For comparison FE model

was constructed using ABAQUS to simulate the experiment. The buckling loads obtained from

the experiment and the FE model were observed to be in good agreement with each other. The

FE model was further used to investigate the cone with non constant thickness.

It was observed that variation of thickness in axial direction results in the reduction of

buckling load. The reduction of buckling load, due to the small thickness variation in axial

direction is proportional to the thickness variation parameter , where is the ratio of the

difference of minimum and maximum wall thickness to the minimum wall thickness of cone.

Huu Tai Thai and Seung Eock Kim have performed the Inelastic post buckling analysis

of space steel trusses using the generalized displacement control method[5]. Space steel

trusses used extensively for domes or roofs. In this paper the authors have extended the

application of Hill [11] model for inelastic post buckling analysis of space steel trusses.

Geometric and material non linearity are considered for the study.


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This paper presents an algorithm that can trace the equilibrium paths of the non linear

problem with multiple limit points and snap back points. A computer program is developed to

predict the inelastic post buckling behavior of space truss structures. The paper includes a

number of examples solved to prove the accuracy of the proposed procedure.

A.B. Sabir and A. C. Lock have studied the application of Finite Element

Methods to the Large Deflection Geometrically Non Linear behavior of Cylindrical Shells[9]. The

paper presents a method that is capable of dealing with the study of post buckling of structures

subjected to in plane and lateral loading. It employs linearized incremental method and Newton

Raphson method.

The algorithm based on the linearized incremental and Newton Raphson

methods, has been used on cylindrical shells subjected to lateral loading and a flat plate

subjected to in plane loading. A family of shells of similar dimensions but varying thickness has

been examined and load deflection curves exhibiting complicated behavior are obtained.

H Ramsey has studied the plastic buckling of conical shells under axial compression

[6]. This research paper presents analysis and experimental results of plastic axisymmetric

buckling of steep, truncated conical shells subjected to axial compression. For this, specimens

of 6061-T6 aluminum and stainless steel were tested by subjecting to compressive load. In spite

of the remarkable difference in the stress-strain curve for these materials, same buckling modes

- were observed for the geometrically identical cones. The analysis uses perturbation method,

wherein no minimum load is required for initiation of buckling, but, the plastic zone must be

sufficiently developed for the formation of preferred buckling mode. The analysis method

described does not predict the maximum load, but gives good description of the buckling

deformation.

N K. Gupta and H. Abbas have studied the axisymmetric axial crushing of thin

frusta[7]. It is normally observed in frusta with semi-epical angles


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convolution. A mathematical model for the calculation of variation of crushing load is developed.

Experiments were performed by subjecting Aluminum frusta to axial compression. During

experiments it was observed that the limb of the first fold is straight as compared to other folds

thus indicating less absorption of flexural energy in it. The fold size and crushing load are in

good agreement with the experimental results.

A Spagnoli, M. K. Chryssanthopoulos have studied the Elastic buckling and

post buckling behavior of widely stiffened conical shells under axial compression[8]. In this

paper, the linear and non-linear elastic buckling response of the conical panel is studied for a

wide range of shell and stiffening parameters by means of an appropriate finite element model.

Linear analysis is used for the determination of classical buckling load. The non-linear analysis

of imperfect conical shells is used for the study of imperfection sensitivity. The FE package

ABAQUS is used for modeling and analysis.

B S Golzan and H. Showkati have studied the buckling of thin walled conical

shells under uniform external pressure[10]. This paper presents a detail study of the facility

developed for the buckling experiments on conical shells. The experiments were performed

using six test specimens which were loaded by applying external pressure, and the respective

buckling load and buckling modes were studied. The experimental buckling loads obtained were

compared with the ones derived from FEA and Jawad equation.The experimental buckling load

is observed to be lower that obtained from FEA and the Jawad equation.

H. A. Mang, G Hofinger and X. Jia have studied the Sensitivity Analysis of the Post Buckling

Behavior of Elastic Structures[14]. This paper presents a study that focuses on the sensitivity

analysis and the conversion of imperfection sensitive structures to imperfection insensitive

structures. Several examples are presented that focus on special cases of buckling.


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CHAPTER 3

BUCKLING AND POST BUCKLING

3.1 Buckling

Buckling is that mode of failure when the structure experiences sudden failure when

subjected to compressive stress. When a slender structure is loaded in compression, for small

loads it deforms with hardly any noticeable change in the geometry and load carrying capacity.

At the point of critical load value, the structure suddenly experiences a large deformation and

may lose its ability to carry load. This stage is the buckling stage.

The critical buckling load for a pin-pinned column is given by the formula

2

2

l

EIP

cr

= Eq. 3.1

The structural instability of buckling can be categorized as

Bifurcation buckling

Limit load buckling

In Bifurcation buckling the deflection when subjected to compressive load, changes

from one direction to a different one. The load at which bifurcation occurs is the Critical Buckling

Load. The deflection path that occurs prior to the bifurcation is called as the Primary Path and

that after bifurcation is called as secondary or post buckling path.

There are two types of buckling failure problems in thin shells: Axis-symmetric problems and

non axis-symmetric problems. Axis-symmetric problems refer to problems in thin axis-symmetric

shells subjected to axis-symmetric load. For these structures instability may occur in the form of

axis-symmetric snap through, or non axis-symmetric bifurcation.

3.2 Post Buckling

Post buckling stage is a continuation of the buckling stage. After the load reaches its

critical value the load value may not change or it may start decreasing, while deformation

continues to increase. In some cases the structure continues to take more load after certain


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3.2.2 Material Non linearity

Material nonlinearities arise from the presence of time independent behavior such as

plasticity, time dependent behavior such as creep and viscoelastic / viscoplastic behavior where

both plasticity and creep effects occur simultaneously. These may result in irreversible structural

behavior. Non-linear behavior in ANSYS is characterized as,

Plasticity- permanent time independent deformation.

Creep: Permanent, time dependent deformation.

Non-linear elastic: non linear stress strain curve, structure returns to original state on unloading,

no permanent deformation.

Viscoelasticity: Time dependent deformation under constant load. Structure Returns to original

state upon unloading.

Hyper elasticity: Rubber - like materials.

3.2.3 Contact Non linearity

Nonlinearity due to contact conditions arises because the prescribed displacements on

the boundary depend on the deformation of the structure.

This paper focuses on the study of buckling and post buckling of various structural members

using Finite element software ANSYS APDL and ANSYS Workbench 12.1.

3.3 Finite Element method and ANSYS

The finite element method (FEM) is a numerical technique for finding approximate

solutions of partial differential equations (PDE) as well as of integral equations. The solution

approach is based either on eliminating the differential equation completely (steady state

problems), or rendering the PDE into an approximating system of ordinary differential equations,

which are then numerically integrated.

Finite Element Analysis (FEA) is a computer simulation technique used in engineering analysis

by using the numerical technique of finite element method (FEM). One mechanical engineering

software widely used for such analysis is ANSYS.


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ANSYS structural mechanics offers wide range of analysis from concept simulation to

advanced analysis. With a full complement of linear and nonlinear elements, material laws

ranging from metal to rubber and the most comprehensive set of solvers available, ANSYS

simulation tools are applied widely by users across industries. Additionally, the adaptive

architecture of ANSYS software tools provides one with the flexibility for customization and

interoperability with other tools such as third-party software.


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CHAPTER 4

TRUSS AND ARCH STRUCTURES

Trusses find wide range of applications in structures such as bridges, roof structures,

railways, etc. The arch is significant because when subjected to vertical loads, its two ends

develop reactions inwardly in horizontal direction. The arch can carry a much greater load than

a flat beam of the same size and material, because downward pressure forces the elements

together instead of apart. Thus, these structures are expected to sustain heavy loads.

The Finite element analysis is used for the study of post buckling behavior of these

structural components. The method is implemented by the use of softwares like ANSYS APDL

and ANSYS Workbench.

4.1 Investigation Process

In order to analyze the structural components using ANSYS the following process is

proposed:

The first step is to create a simplified model of the structure to be analyzed;

The second step is to apply material properties and the element type and create a mesh of

appropriate density;

The third step is to apply defined boundary conditions and set up the non-linear solution

controls and solve the problem to give the post buckling results.

4.2 ANSYS Simulation

This work includes the buckling and post buckling analysis of several structural

components. These include Shallow Truss, Shallow Arch, Diagonal Truss, Cylindrical Panel and

Conical Frustum.


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4.2.1 Shallow Truss

A stable truss design exists when deformations increase as the applied load increases;

an unstable design occurs when deformations increase as the load decreases. To understand

this type of structural response, the equilibrium paths of key connection points are plotted. The

shallow truss geometry prompts snap-through buckling of many members rather than local

buckling of a single member.

The Shallow Truss is analyzed here by subjecting it to two different sets of Boundary

conditions;

- Clamped

- Hinged

4.2.1.1 ANSYS Simulation

a) Geometry: The Shallow Truss geometry is shown in the following Figure 4.1

Figure 4.1 Shallow Truss geometry

H1

W

H2

0.753 in

0.243 in


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The dimensions of the Truss are as follows,

Table 4.1 Dimensions of Shallow Truss

H1 25.886 in

H2 12.943 in

W1 0.386 In

b) Material Properties

ANSYS offers a wide range of material properties to be used for a particular analysis.

For the truss, the material is assumed to be homogenous linear isotropic, hence the only

properties needed are;

Table 4.2 Material Properties for Shallow Truss

Youngs Modulus 10.3 106

lb/in2

c) Boundary Conditions

- Clamped

For the clamped truss, the mesh is formed by using BEAM3 elements. Beam3 is a

uniaxial element with tension, compression, and bending capabilities [18].

Figure 4.2 BEAM3 element


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The element has three degrees of freedom at each node: translations in the element X

and Y directions and rotation about the element z-axis. The element is defined by two nodes,

the cross-sectional area, the area moment of inertia, the height, and the material properties and

has special features like large deflection [13].

The truss is analyzed by discretizing the domain into different number of elements and

the mesh that gave a satisfactory solution is implemented. After a number of trials the

appropriate number of elements found to give favorable result is 6.

For constraints, the structure is fixed in all degrees of freedom at both ends and a

displacement of 0.9 in is applied in the negative Y direction at the center node.

When a displacement is applied at the central node of the truss, the node is pushed in

the downward direction creating a reaction force. This reaction force represents the force that

will be needed to achieve the particular amount of displacement.

Figure 4.3 Clamped Truss subjected to Boundary conditions

Dis lacement=-0.9 in

FixedFixed


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- Hinged

For the truss with hinged boundary conditions, element type used for the mesh is LINK1.

Figure 4.4 LINK1 element

The element is defined by two nodes, the cross-sectional area, an initial strain, and the

material properties with features like plasticity, large deflection, birth and death etc. As in a pin-

jointed structure, no bending of the element is considered [13].

In this case, the truss is hinged at both ends i.e. displacements in X and Y directions are

fixed and a displacement of 0.9 in the negative Y direction is applied at the central node.

Figure 4.5 Truss subjected to hinged boundary conditions

Hinge Hinge

Dis lacement=0.


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4.2.1.2 Results

The equilibrium plots of the above analyzed structures are plotted in figure 4.7 and 4.10.

The nonlinear instability region is the region in which snap-through occurs and in which the

equilibrium path goes from one stable point (a) to another new stable point (b). The nonlinear

behavior places the critical limit load at point (a) equal to that at point (b), but the load limit

corresponds to a new structural shape. In case of bifurcation buckling at bifurcation point the

structure immediately becomes unstable and buckles. The member is unable to support any

additional load, which is not the case for nonlinear snap-through buckling.

The above cases are analyzed in ANSYS APDL. After subsequent trials a mesh of 6

elements is used.

a) Clamped Shallow Truss

For the clamped shallow truss, the resulting deformation is shown in figure 4.6.

Figure 4.6 Deformed clamped truss

The graph of vertical load reaction at the central node against the displacement is

plotted in the figure 4.7. The results match with the results in [3]. Vertical load reaction is the


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force reaction that is evaluated at the point where displacement is applied. The analytical limit

point for the truss is obtained at a load value of 38 lb and deflection of 0.22 in. In reference [3]

the limit point occurs at 33.3 lb and deflection of 0.2 in.

Figure 4.7 Load (lb) Vs displacement (in)

a

b


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Figure 4.8 Load (lb) Vs Horizontal reaction

The maximum compressive force obtained in the members is 750 lb. In this case, as the central

node is gradually pushed downward, the vertical force reaction goes on increasing till it reaches

point (2) and so does the horizontal reaction force evaluated at the truss ends. The vertical load

values in (1-2-1) region of the graph in Figure (4.8) correspond to those in the post-buckling

region in Figure (4.7). The horizontal reaction goes on decreasing after it reaches point (2).

b) Hinged Shallow Truss

The shallow truss with hinged boundary conditions is deformed as depicted in the Figure (4.9)

1

2


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Figure 4.9 Deformed hinged truss

The deformation of the central node is plotted against the force reaction at that node

and the observed trend is plotted in the following Figure 4.10. The graph displays the post

buckling behavior of the structure. After the first peak of load, the displacement is still increasing

but the evaluated force reaction is reducing till the next negative peak is reached and starts

increasing again. The result matches with the results in [3].


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Figure 4.10 Load (lb) Vs deflection (in)

The limit point (a) for the truss occurs at 18 lb and 0.16 in.

The horizontal reaction force evaluated at the ends of the truss is plotted against the vertical

load reaction in Figure 4.11. The maximum compressive force obtained is 850 lb. After reaching

the maximum value at point (1) horizontal force decreases gradually. At point (1) the vertical

force becomes zero and then continues to be in the negative range till it again attains the value

of zero at point (2). The results match with the results in [3]. Reference [3] utilizes a 5 element

mesh for both cases to analyze the post buckling behavior of structures.

ab


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Figure 4.12 Diagonal truss Geometry

The dimensions used to model the truss are given in the following table.

Table 4.3 Diagonal Truss Dimensions

H1 48 in

V1 36 in

Cross sectional Area 1 in


A bilinear isotropic material is defined for the truss;

H1

V1


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Table 4.4 Material Properties for Diagonal Truss

Youngs Modulus 3107

psi 2.071011

Pa

Yield Stress 30,000 psi 2.0710

8

Pa

Tangent Modulus 3105

psi 2.07109

Pa

Figure 4.13 Stress Strain Curve

The tangent modulus is useful in describing the behavior of materials that have been

stressed beyond the elastic region. When a material is plastically deformed, initially the

relationship between stress and strain is linear with a high slope, and later after point (1) is

reached, the slope of the line changes to a smaller value. The tangent modulus quantifies the

"softening" of material that generally occurs when it begins to yield. Although the material

softens, it is still generally able to sustain more load before ultimate failure. Therefore, more

weight efficient structure can be designed when plastic behavior is considered, hence in

structural analysis the tangent modulus is used to quantify the buckling failure [19].


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A displacement of 3 in is applied on node 6 in the negative Y direction. Nodes 9 and 1 are fixed

as shown in figure 4.13.

Figure 4.14 Boundary conditions for Diagonal Truss

4.2.2.2 Results

Due to the applied boundary conditions the truss is deformed. The deformed shape of the truss

is as shown in the Figure 4.14.

Node 9

Node 1


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Figure 4.15 Deformed Shape of Diagonal Truss

Following is the plot of load reaction Vs deflection in Figure 4.16. The limit point of the truss

occurs at a load of 1800 lb and a deflection value of 0.2 in. When the applied displacement

pushes the truss in the negative Y direction, buckling is experienced in members 1 and 2

leading to a plastic deformation of the truss. The results match with the reference [15], where a

non linear analysis program ADINA is used.

1

2


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Figure 4.16 Load (lb) Vs Deflection (in)

4.2.3 Shallow Arch

4.2.3.1 ANSYS Simulation

The shallow arch is modeled using BEAM3 elements.

a) Geometry

The shallow arch is modeled using the following dimensions,

Table 4.5 Dimensions of Shallow Arch

Cross Section Area 0.188 in

Area Moment of Inertia 0.00055 in4

H 1.09 in

H 0.1875 in

L 34 in


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Figure 4.17 Shallow Arch geometry


The material is assumed to be linearly elastic with the following properties,

Table 4.6 Material Properties of Shallow Arch

Youngs Modulus 10106lb/in

Poissons Ratio 0.2


The arch is fixed at both ends for all degrees of freedom and a displacement of 1.8 in, in the

negative Y direction is applied at the central node.

L

hH

1 in

0.1875 in


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Figure 4.18 Boundary Conditions for Shallow Arch

4.2.3.2 Results

Mesh convergence is obtained on discretizing the domain by employing 24 elements. The

applied boundary conditions cause the arch to deform at the center and towards the ends as

shown in the figure 4.19 and 4.20 attached below.

Fixed Fixed

Dis lacement=-1.8 in


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Figure 4.19 Deformed Shape of Shallow Arch with central displacement = -1.8 in

Figure 4.20 Deformed Shape of Shallow Arch with central displacement = -0.5 in


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The vertical load reaction at the central node is plotted against the deflection at the central

node,

Figure 4.21 Load (lb) Vs Displacement (in)

The limit point of the truss occurs at a load value of 35 lb and a deflection value of 0.4 in.

As the point of critical load is reached the structure goes into the non linear i.e. post buckling

region. The region continues till the load of 35 lb is reached at a corresponding displacement

value of 1.75 in.

The results match with those given in reference [15], where half of the arch is analyzed

using 12 equal length beam elements.


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CHAPTER 5

SHELL STRUCTURES

5.1 Cylindrical panel

The cylindrical panel is meshed by employing Shell 181 elements. Shell 181 is a 4-node

element with six degrees of freedom at each node: translations in the x, y, and z directions, and

rotations about the x, y, and z-axes. Its features include plasticity, elasticity and large deflection

[18].

Figure 5.1 Element Shell 181

5.1.1 ANSYS Simulation

a) Geometry: The panel is a part of a cylinder that has the following dimensions,

Table 5.1 Dimensions of Panel

Radius, R 2540 mm

Length 580 mm

Thickness 12.7 mm

Included angle of panel 0.2 radian


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Figure 5.2 Geometry of Cylindrical panel

Due to symmetry a quarter portion of the panel is modeled and analyzed. The geometry of the

modeled part and the mesh density is shown in the following figure 5.3.

Figure 5.3 Mesh of Cylindrical Panel

m


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The panel possesses linear elastic material properties:

Table 5.2 Material Properties of panel

Youngs Modulus 3102.75 MPa

Poissons Ratio 0.3


Symmetry boundary conditions are applied on the inner edges. The cylindrical panel is

simply supported on the outer edges. The problem is solved for the following cases of loading

on the central node,

c.i) Applied displacement of 100 mm

c.ii) Applied force of 1000 N

Figure 5.4 Applied Displacement at central Node

Dis lacement


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When displacement is applied at the center of the panel, the panel experiences a

reaction force against the applied displacement. The reaction force at the central node is

evaluated as shown in Figure 5.7,

Figure 5.7 Force Reaction at Central Node

The equilibrium path for the panel is plotted below in Figure 5.7. After the limit load, the

path enters the non-linear region and gradually approaches the new stable point which would

have the critical load equal to the first one but corresponding to a new structural shape.


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Figure 5.8 Load Vs Displacement

From the above results we observe that the limit point for the panel occurs at a load of

556.59 N. Since, a quarter portion of the panel is considered, the actual limit point for the whole

panel occurs at 2226 N.

These results are in agreement with the results in reference [3] which uses a 16 element mesh

for the quarter model.

When a force is applied at the central node, a maximum displacement of 30.7 mm in

the negative Y direction occurs at the central node of the panel. The deformed shape of the

panel is shown in figure 5.8.


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Figure 5.9 Deformed Shape of panel due to applied load

5.2 Conical Frustrum

Conical frusta find wide applications as energy absorbers. Hence, buckling and post

buckling is an important area of study for these.

The buckling and post buckling behavior of thin walled conical frusta subjected to axial

compression is analyzed. The component is assumed to be subjected to geometric and material

non-linearity. The conical frusta in the following problem are simulated by crushing them

between rigid plates and hence experience contact non linearity [2].

5.2.1 ANSYS Simulation

a) Geometry

A number of conical specimens were analyzed using ANSYS Workbench. Following are the

dimensions of the specimens,


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Figure 5.10 Geometry of Frustum

Table 5.3 Dimensions of Conical frusta

Specimen () Db Dt t Hc

C6 18 135 88 0.9 68

C25 23 165 89 0.65 88.1

C25 23 165 89 0.7 88.1

C18 24 152 90 0.9 68

C2 16 128 88 0.9 69

All dimensions are in mm

The frustum and the plate are meshed using the Shell 181 element. While performing

the analysis the plate on top of the cone is assumed to be rigid. Hence, when the 3D assembly

is meshed, for the plate, a single mass element is created at its centroid. For such cases the

mesh is created only on the faces in the defined contact [16].

Dt

Db

Hc

t

2


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Figure 5.11 Mesh and geometry of specimen C6


The material assumed for this problem is Aluminum.

Table 5.4 Material Properties of Frusta

Youngs Modulus 70 GN/m

Poissons Ratio 0.35

Yield Stress 55 MPa

Tangent Modulus 4213 N/mm

The plate is made of structural steel and assumed to possess linear material properties,

Table 5.5 Material Properties of Rigid Plate

Youngs Modulus 200 GN/m

Poissons Ratio 0.3


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The cone is subjected to axial compression between rigid platens. Hence, while

modeling, the bottom edge of the cone is fixed and the top of the cone is subjected to

compression by the application of remote displacement of 15 mm on the rigid plate. With remote

displacement, workbench applies the specified displacement to the geometric center of the rigid

body.

Figure 5.12 Boundary conditions applied to specimen C6

5.2.2 Results

The applied displacement causes the conical frustum to deform. The deformed shape of the

frustum is shown in the figure 5.13.


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Figure 5.13 Deformed Shape of frustum

Figure 5.14 Reaction Force evaluated at contact region


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The Force Vs displacement graph for specimen C6 is shown in fig (5.14). Aluminum

represents a ductile type of failure. The peak load observed is 9.3 kN for a given displacement

of 15 mm. The initial peak is observed for load value of 7.1 kN and then the load drops to 4.3 kN

which is then followed by a rising trend.

Figure 5.15 Load Vs Displacement

The results obtained from the analysis in ANSYS workbench by using shell 181 element

are fairy comparable with the ones in [2]. The reference uses element type Shell 143 to mesh

the frusta and the software used to accomplish the analysis is ANSYS APDL.


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Table 5.6 Force Values for Frusta

ANSYS Workbench ANSYS APDL Experimental

Element Type Shell 181 Shell 143

SpecimenMax Force

(kN)

Avg Force

(kN)

Max Force

(kN)

Avg Force

(kN)

Max Force

(kN)

C6 9.3 5.9 8.2 5.5 3.4

C25(0.65) 5.8 3.1 4.4 13.7 4.9

C25 (0.7) 6.3 3.5 4.4 13.7 4.9

C18 7.7 5.1 12.7 8.35 5.5

C2 14.7 7.7 9.2 7.2 4

The observed differences in the force values are the cumulative result of the difference

in Mesh density and the element type implemented. The Mesh used in this work has larger

mesh density. Also, under bending loads, for Shell 143, inferior results can be produced leading

to a need of a refined mesh [13] [18].

5.3 Eigen Value Buckling Load

The Eigen value buckling analysis is performed for all cases of structures and the Eigen

value buckling load is compared to the limit load obtained from the non-linear buckling

simulation. From the results in Table 5.7 it is observed that the eigen value buckling load is

higher in each case.

Eigen value buckling load is the load at which the structure experiences instability and

fails immediately.


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Table 5.7 Eigen Value Buckling Load compared to the Non-linear Buckling Limit Load

Structural

ComponentSoftware

Non-Linear Buckling

Limit Load

Eigen-value

Buckling Load

Shallow Truss -

ClampedANSYS APDL 33.3 lb 93.5 lb

Shallow Truss -

HingedANSYS APDL 18 lb 35.8 lb

Diagonal Truss ANSYS APDL 18 kN 26 kN

Shallow Arch ANSYS APDL 35 lb 53.713 lb

Cylindrical Panel ANSYS Workbench 2.2 kN 2.7 kN

Conical frustum

Specimen C6ANSYS Workbench 9.3 kN 17.84 kN

Conical frustum


Conical frustum


Conical frustum


In case of Non-linear buckling analysis, limit load is the critical load after which the structure

goes into a post-buckling stage, from one stable point to other, both havingequal critical load

values but different structural shapes. Hence Eigen value is predicted by ANSYS is higher.


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CHAPTER 6

CONCLUSION AND RECOMMENDATIONS

The main objective of the thesis is to investigate the buckling and post buckling

behavior of structural components that are bound to be subjected to heavy loads. Their complex

equilibrium paths are explained to inform engineers of possible nonlinear behavior in designs

and that instability may occur before a design bifurcation limit is reached. Understanding the

large elastic displacement of these types of structures can prevent sudden buckling failures

from applied operational and construction loads.

Structural components like the shallow truss, shallow arch, diagonal truss and shells

like conical frusta and cylindrical panel are chosen for examination. These are realistic

configurations that are directly usable in practical structural designs. These are simple metallic

structural components from which buckling and post buckling can be studied.

The investigated models displayed snap-through behavior after reaching a critical point.

The nonlinear behavior was identified by plotting the equilibrium paths of the center nodes. Post

buckling region is that area of these equilibrium plots, that lies between the two stable points

that correspond to the same critical load value but different displacement values.

This behavior of the analyzed structures signifies that under loaded condition the

structure does not become unstable and buckle immediately. These load values are compared

to the Eigen buckling load values and Eigen Value buckling load is found to be higher. At this

load the structure becomes unstable and buckles immediately.

The post buckling study can be extended to micro sized structures that are used in micro

electromechanical systems, for example, miniature trusses can be used as switches that can be

integrated into circuits. Also, sensitivity analysis of the post buckling behavior of these

structures can be accomplished. The Finite Element Methods software ANSYS APDL and

workbench can be successfully implemented for the same.


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APPENDIX A

DEFORMED SPECIMENS OFCONICAL FRUSTA


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Figure A.1 Example of Mesh density for frusta in reference [2]

Figure A.2 C25 Deformed Shape for thickness of 0.7 mm


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Figure A.3 C18 Deformed Shape

Figure A.4 C2 Deformed Shape


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APPENDIX B

EIGEN VALUE BUCKLINGMODE SHAPES


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Figure B.1 Shallow Truss Clamped

Figure B.2 Shallow Truss - Hinged


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Figure B.3 Diagonal Truss

Figure B.4 Shallow Arch


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Figure B.5 Cylindrical Panel

Figure B.6 Conical Frustum Specimen C6


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REFERENCES

1] Buckling and post-buckling behavior of thin walled cylindrical steel shells with varying

thickness subjected to uniform external pressure; S Aghajari, K Abedi and H Showkati, Thin

Walled Structures, 44, (2006) 904-909

2] Buckling of thin conical frustum under axial loads; N. K. Gupta, N. Mohamed Sheriff, R.

Velmurugan, Thin Walled Structures, 44, (2006) 986-996

3] Post buckling behavior of plates and shells using a mindlin shallow shell formulation; A. Pica

and R. D. Wood, Computers and Structures, Vol. 12, pp 759-768

4] Buckling of Axially compressed conical shells of linearly variable thickness using structural

model; Chawalit Thinvongpituk and Pisit Techarungpaisarn, The 17th

annual conference of

Mechanical Engineering Network of Thiland, Oct 2003

5] The Inelastic post buckling analysis of space steel trusses using the generalized

displacement control method; Huu Tai Thai and Seung Eock Kim, 5th

International Symposium

on Steel Structures

6] Plastic buckling of conical shells under axial compression; H Ramsey, J. of Mechanical

Science, Vol 19, pp 257-270, 1977

7] Axisymmetric axial crushing of thin frusta; N. K. Gupta and H. Abbas, Thin Walled Structures,

36 (2000) 169-179

8] Elastic buckling and post buckling behavior of widely stiffened conical shells under axial

compression; A. Spagnoli, M. K. Chryssanthopoulos, Engineering Structures 21 (1999) 845-855

9] The application of Finite Element Methods to the Large Deflection Geometrically Non Linear

behavior of Cylindrical Shells, A.B. Sabir and A. C. Lock, Variational methods in Engineering, pp

66-75.

10] Buckling of thin walled conical shells under uniform external pressure, B S Golzan and H.

Showkati, Thin Walled Structures 46 (2008) 516-529


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11] Hill C.D., Blandford G.E. and Wang S.T. (1989) Post-buckling analysis of steel space

trusses, Journal of Structural Engineering, Vol. 115, No. 4, pp. 900-919.

12] Non-Linear Buckling and post buckling analysis of shells of revolution; Tao Hong, Thesis,

Sep 1999

13] www.kxcad.net

14] Sensitivity Analysis of the Post Buckling Behavior of Elastic Structures; H. A. Mang, G

Hofinger and X. Jia

15] Solutions for Structural problems solved using ADINA.

16] Finite Element Simulations with ANSYS Workbench 12

17] ANSYS tutotial: University of Alberta

18] ANSYS Help Manual

19] The Finite Element Method and Applications in Engineering using ANSYS, Madenci,

Erdogan.


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BIOGRAPHICAL INFORMATION

Shruti Deshpande has completed her Bachelors degree in Production Engineering from Pune

university in 2005. She worked with BOSCH for two years after earning the degree. She started

her Masters in Mechanical Engineering in the Spring of 2009 and completed her degree in Fall

2010. During her tenure as a Masters student, she served as a Graduate teaching Assistant for

Dr. Kent Lawrence. Her thesis is based on the study of buckling and post buckling of Structural

Components, incorporating the use of Finite Element Analysis tools such as ANSYS APDL and

ANSYS Workbench. Her work interests are Finite Element Methods and Analysis and Computer

Aided Design.

Documents

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